Bond Pricing and Yield

Module-16
Session-32
Bond Pricing and Yield
The value of bonds can be described in terms of rupee values or the rates of return they
promise under some set of assumptions. The major model used for valuation of bond is the
present value model, which computes a specific value for the bond using a single discount
value.

32.1 Valuation of Bond (Present Value Model)
The value of the bond is equal to the present value of the cash flows expected from it.
For determining the bond price: It requires (i) an estimate of expected cash flow and (ii) an
estimate of required rate of return. The cash flows from a bond are the periodic interest
payments to the bondholder and the repayment of principal at the maturity of the bond.
Therefore the value of a bond is the present value of the annual or semi annual interest
payments plus the present value of the principal payment
Bond value with annual interest payment:
Pm =
Where:

n

Ct

∑ (1 + i )
t =1

t

+

Pp
(1 + i ) n

Pm=the current market price of the bond
n = the number of years to maturity
Ci = the annual coupon payment for bond i
i = the prevailing yield to maturity for this bond issue
Pp=the par value of the bond
P= Ct*PVIFA r, n + Pp *PVIFr,n
P= Value of the bond, PVIFA = Sum of an annuity of rupee 1 per period for n periods, Pp =
par value of the bond and PVIF = Present value of rupee one
Bond values with Semi Annual Interest
2n

Pm = ∑
t =1

Pp
Ct 2
+
(1 + i 2) t (1 + i 2) 2 n

Where:
Pm=the current market price of the bond
n = the number of years to maturity
1

Ci = the annual coupon payment for bond
i = the prevailing yield to maturity for this bond issue
Pp=the par value of the bond
P= Ct/2*PVIFA r/2, 2n + Pp *PVIFr/2,2n

31.2 Bond Yield Measures
The investor in bond typically receives income from the following:
I. Internet payments at a contracted rate i.e. coupon interest.
II. Capital gain or loss arising out of sale of the bond.
III. Cash realization on sale of bond.
IV. Redemption of the bond by the issuers at a contracted value.
Items (i) and (ii) constitute returns to the bond investor, while (iii) and (iv) are principal
recoveries. An investor’s income on bond investments depends on whether he holds the
bond to maturity or disinvests before maturity. If the bond is held to maturity, cash flows
(i) and (ii) will accrue. However, if he sells before maturity he receives cash flows (i), (ii)
and (iii) above. The return to the bond investor can be measured in terms of the
following:
(a.) Current Yield (CY)
(b.) Yield to Maturity (YTM)
(c.) Realised Yield (RY)
(d) Yield to Call (YTC)
Current Yield (CY)
CY is measured by comparing (i) with the prevailing market price. Thus,
CY = coupon interest/ prevailing market price
Because this yield measures the current income from the bond as a percentage of its price, it
is important to income-oriented investors who want current cash flow from their investment
portfolios. An example of such an investor would be a retired person who lives on this
investment income. Current yield has little use for investors who are interested in total return
because it excludes the important capital gain or loss component.
Example:
A 8% bond (Face value of Rs. 100) selling for Rs.96, would have a current yield of,
Rc = 8/96 = 8.33%

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Current yield of bonds selling at par would be equal to the coupon interest rate.
Current yield of bonds selling at a premium (discount) would be less (more) than the coupon
interest rate. An important drawback of current yield is that it considers only coupon income
as a source of return to the investor, ignoring interest and capital gain (loss) that would also
accrue to him.
Yield to maturity (YTM)
The correct way of computing the return on any asset involve considering the entire
sequence of cash flows and their timing and calculating the Internal Rate of Return (IRR). In
the case of a bond, there is a cash outflow (equal to the price of the bond) when the bond is
bought and there are cash inflows when the periodic interest coupons are received and when
the redemption value is received on maturity. Calculating the IRR of this stream of cash
flows gives the true returns on the bond, which is known as Yield-To-Maturity (YTM). Thus,
yield-to-maturity is measured by comparing the present values of (i) and (iv) dealt in the
previous paragraphs with the prevailing market price.
2n

Pm = ∑
t =1

Pp
Ci 2
+
t
(1 + i 2) (1 + i 2) 2 n

To compute the YTM for a bond, we solve for the rate i that will equate the current
price (Pm) to all cash flows from the bond to maturity. As noted, this resembles the
computation of the internal rate of return (IRR) on an investment project. Because it is a
present value–based computation, it implies a reinvestment rate assumption because it
discounts the cash flows. That is, the equation assumes that all interim cash flows (interest
payments) are reinvested at the computed YTM. In this method of calculation we assume that
(i) Investor holds bond to maturity
(ii) All coupon and principal payments are made as per the schedule
(iii)All the bond’s cash flow is reinvested at the computed yield to maturity
The impact of the reinvestment assumption (i.e., the interest-on-interest earnings) on
the actual return from a bond varies directly with the bond’s coupon and maturity. A higher
coupon and/or a longer term to maturity will increase the loss in value from failure to reinvest
the coupon cash flow at the YTM. Put another way, a higher coupon or a longer maturity
makes the reinvestment assumption more important. The yield to maturity will be calculated
in the above-mentioned formula by trial and error. Some of the investors calculate the YTM
by an approximation method.

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Approximation to YTM
As we know total return of the bond consists of interest payments and capital gain /
loss on redemption. Therefore, the average annual return can be calculated as:
Average annual return = [C + (F-P) / n] / [(F+P)/2]
Where, C = coupon, F =redemption value and P = purchase price.
It is also denoted as: Average annual return = Annual interest + [Capital gains / Number of
years].
Similarly one can also calculate the average investment as the average of the current
price and redemption value. Many investors then compute the yield as the average annual
return divided by the average investments. Many other approximations of this kind are in
common use. Some approximations do not use the notions of average investment, but divide
the return either by the redemption value or by the price. There are some approximations
which divide the annual interest by the price and divide the capital gains by the redemption
value. All these approximations involve the same error of ignoring the timing of the cash
flows. These approximations are however; very useful in calculating the YTM by trial and
error since they provide a very good initial guess, the process of trial and error gives the true
YTM in few steps.
Realized Yield (RY)
Realized yield is the yield actually earned by the investor on his or her investment
and depends on the reinvestment rate and holding period chosen by the investor. The realized
yield (i.e., the actual return over a horizon period) measures the expected rate of return of a
bond that you expect to sell prior to its maturity. The realized yield can be stated as the rate
that equates the future value of the purchase price to the total cash flow realized on the bond.
In terms of the equation, the investor has a holding period (hp) or investment horizon that is
less than n. Realized (horizon) yield can be used to estimate rates of return attainable from
various trading strategies.
The approximate realized yield can be calculated as:
ARY = [Ci + (Pf – Pm) / hp] / [Pf + Pm /2]
Where,
ARY = approximate realized yield
C = annual coupon payment of the bond i
P = future selling price of the bond
P = current market price of the bond
hp = holding period of the bond
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Since the realized yield measures are based on an uncertain future selling price, the
approximate realized yield method is appropriate under many circumstances. The substitution
of Pf and hp into the present value model of bond provides the following realized yield model
2 hp

Pm = ∑
t =1

Pf
Ci 2
+
t
(1 + i 2)
(1 + i 2) 2 hp

This above formula represents the standard present value formula with changes in
holding period and ending price. It includes the implicit reinvestment rate assumption that all
cash flows are reinvested at the computed realized yield.

In many cases, this is an

inappropriate assumption because available market rates might be very different from the
computed realized (horizon) yield. Therefore, to derive a realistic estimate of the estimated
realized yield, you also need to estimate your expected reinvestment rate during the
investment horizon.
Yield to Call (YTC)
Whenever a bond with a call feature is selling for a price above par (that is, at a
premium) equal to or greater than its call price, a bond investor should consider valuing the
bond in terms of YTC rather than YTM. This is because the marketplace uses the lowest,
most conservative yield measure in pricing a bond. The yield to call can be measured using
the present value model and in this case the model assumes that you hold the bond until the
first call date and that you reinvest all coupon payments at the YTC rate. To compute the
YTC by the present value method, we would adjust the semi-annual present value equation to
give

2 nc

Pm = ∑
t =1

Ci 2
Pc
+
t
(1 + i 2)
(1 + i 2) 2 nc

Where
Pm = the current market price of the bond
Ci = the annual coupon payment of bond i
nc = the number of years to first call date
Pc = the call price of the bond

32.3 Price –Yield Relationship
•

If yield < coupon rate, bond will be priced at a premium to its par value

•

If yield > coupon rate, bond will be priced at a discount to its par value

•

Price-yield relationship is convex (not a straight line)

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Questions and Answers
1. What do you understand by bond pricing?
Ans. Bond Pricing:
•

The value of the bond is equal to the present value of the cash flows expected from it.

•

For determining the bond price: It requires an estimate of expected cash flow, an
estimate of required rate of return

Bond values with Annual Interest
n

Where:

Pm = ∑
t =1

Pp
Ct
+
t
(1 + i ) (1 + i ) n

Pm=the current market price of the bond
n = the number of years to maturity
Ci = the annual coupon payment for bond i
i = the prevailing yield to maturity for this bond issue
Pp=the par value of the bond
P= Ct*PVIFA r, n + Pp *PVIFr,n
Bond values with Semi Annual Interest
The present-value model
2n

Pm = ∑
t =1

Pp
Ct 2
+
(1 + i 2) t (1 + i 2) 2 n

Where:
Pm=the current market price of the bond
n = the number of years to maturity
Ci = the annual coupon payment for bond i
i = the prevailing yield to maturity for this bond issue
Pp=the par value of the bond
P= Ct/2*PVIFA r/2, 2n + Pp *PVIFr/2,2n
2. Expalin Price –Yield Relationship.

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Ans.
•
•
•

If yield < coupon rate, bond will be priced at a premium to its par value
If yield > coupon rate, bond will be priced at a discount to its par value
Price-yield relationship is convex (not a straight line)

The Yield Model: The expected yield on the bond may be computed from the market price
2n

Pm = ∑
t =1

Pp
Ci 2
+
t
(1 + i 2) (1 + i 2) 2 n

Where,
Where:
i = the discount rate that will discount the cash flows to equal the current market price of the
bond

3.What are the various measures available for computing bond yield?
Ans.

4. Explain the meaning of Yield to Maturity.
Ans.
•
•

Promised Yield to Maturity : Widely used bond yield figure
Assumes: Investor holds bond to maturity, all the bond’s cash flow is reinvested at the
computed yield to maturity
2n

Pm = ∑
t =1

Pp
Ci 2
+
t
(1 + i 2) (1 + i 2) 2 n
7

Solve for i that will equate the current price to all cash flows from the bond to maturity,
similar to IRR
Computing the Promised Yield to Maturity:
•

Approximate promised yield
APY =

Ci +

Pp − Pm

n
Pp + Pm

2
= Coupon + Annual Straight-Line Amortization of Capital Gain or Loss
Average Investment
•

Present-value model:

2n

Pm = ∑
t =1

Pp
Ci 2
+
t
(1 + i 2) (1 + i 2) 2 n

5. Write short note on current yield.
Ans.
Current Yield= CY = Ci/Pm
where:
CY = the current yield on a bond
Ci = the annual coupon payment of bond i
Pm = the current market price of the bond

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